Problems

To each pair of numbers $x$ and $y$ some number $x\ast y$ is placed in correspondence. Find $1993\ast 1935$ if it is known that for any three numbers $x,y,z$, the following identities hold: $x\ast x=0$ and $x\ast (y\ast z)=(x\ast y)+z$.

Definition. Let the function $f(x,y)$ be valid at all points of a plane with integer coordinates. We call a function $f(x,y)$ harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: $$f(x,y)=1/4(f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)).$$ Let $f(x,y)$ and $g(x,y)$ be harmonic functions. Prove that for any $a$ and $b$ the function $af(x,y)+bg(x,y)$ is also harmonic.

Let $f(x,y)$ be a harmonic function. Prove that the functions ${\mathrm{\Delta }}_{x}f(x,y)=f(x+1,y)-f(x,y)$ and ${\mathrm{\Delta }}_{y}f(x,y)=f(x,y+1)-f(x,y)$ will also be harmonic.

Liouville’s discrete theorem. Let $f(x,y)$ be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant $M$ such that $\mathrm{\forall }(x,y)\in {\mathbb{Z}}^2$ $|f(x,y)|\le M$. Prove that the function $f(x,y)$ is equal to a constant.